Problem 62 - Algebra 2

Question

Consider \(f(x)=\frac{-15 x^{2}+10 x}{5 x}\) and \(g(x)=-3 x+2\). How can you use what you have observed with \(f(x)\) and \(g(x)\) to verify that expressions are equivalent or to identify excluded values?

Step-by-Step Solution

Answer

The functions are equivalent with an excluded value of \( x = 0 \).

Step 1: Simplify the Function f(x)

Begin by simplifying the function \( f(x) = \frac{-15 x^2 + 10 x}{5 x} \). Factor the numerator to simplify the fraction.The numerator is \(-15x^2 + 10x\). Factor out a common factor of \(-5x\):\[-15x^2 + 10x = -5x(3x - 2)\]So, \( f(x) \) becomes:\[f(x) = \frac{-5x(3x - 2)}{5x}\]Cancel the \(5x\) terms in the numerator and denominator to get:\[f(x) = -(3x - 2)\] which simplifies to \(f(x) = -3x + 2\).

Step 2: Compare with g(x)

The function \( g(x) \) is given as \(-3x + 2\). After simplification of \( f(x) \), we found that \( f(x) = -3x + 2 \) as well.Therefore, \( f(x) \) is equivalent to \( g(x) \).

Step 3: Identify Excluded Values

When simplifying \( f(x) = \frac{-15 x^2 + 10 x}{5 x} \), the division by \( x \) introduces the possibility of excluded values for which the original expression is undefined.The denominator of \( f(x) \) is \( 5x \), which is zero when \( x = 0 \).Thus, \( x = 0 \) is an excluded value because it makes the original equation undefined, even though it does not affect the equivalence with \( g(x) \) after simplification.

Key Concepts

Simplifying Expressions

In mathematics, simplifying an expression is all about making it easier to work with by reducing its complexity. For example, let's look at the function \( f(x) = \frac{-15x^2 + 10x}{5x} \). The first step is to simplify the numerator, \(-15x^2 + 10x\), by factoring. We notice that \(-5x\) is a common factor. After factoring, we get \(-5x(3x - 2)\).

This allows us to simplify \( f(x) \) by canceling out the common factors present in both the numerator and the denominator. Here, \(5x\) can be cancelled, resulting in the expression \(-(3x - 2)\), which simplifies to \(-3x + 2\).

By simplifying expressions, you can perform operations more easily and understand the behavior of functions with less algebraic clutter.

Equivalent Expressions

Equivalent expressions are different expressions that represent the same value for all legitimate variable inputs. In our exercise, we are given \( g(x) = -3x + 2 \) and have found that \( f(x) \), after simplification, is also \(-3x + 2\).

To confirm that two expressions are equivalent like \( f(x) \) and \( g(x) \), you can simplify one of them and see if it transforms into the other. Since both expressions are equal regardless of the input \( x \) (except for any excluded values), they are considered equivalent expressions.

This concept is fundamental because it allows us to replace complex expressions with simpler ones, making problem-solving more efficient and easier to understand.

Excluded Values

While simplifying expressions, one might sometimes encounter values that make the original expression undefined. These are known as excluded values. For the function \( f(x) = \frac{-15x^2 + 10x}{5x} \), consideration of the denominator is crucial.

The denominator \( 5x \) becomes zero when \( x = 0 \). Division by zero is undefined in mathematics; therefore, \( x = 0 \) is an excluded value for the original function.

Identifying excluded values is a critical step when working with expressions because it ensures that we don't make invalid calculations or assumptions about a function's equivalence over all inputs.